1. Field of the Invention
This invention relates to a dynamical system analyser, and more particularly to such a device applicable to analysis of dynamical systems which might be nonlinear or chaotic in the mathematical sense.
2. Discussion of Prior Art
Investigation of nonlinear dynamical systems presents particular difficulties because traditional electronic spectral analysis tools are not appropriate. As an example, consider a chaotic nonlinear dynamical system arranged to produce an electronic signal characteristic of its behaviour. It is known from dynamical systems research that such a signal can have a substantial continuous component to its spectrum. Filtering in either time or frequency to improve signal to noise ratios can distort or alter the system dynamics as perceived from a filtered signal. This is exemplified by the papers of Badii et al in Phys. Rev. Lett. 60 (1988) p.979 and Milschske et al. in Phys. Rev. A37 (1988) p. 4518. Moreover, power spectrum analysis is insufficient to characterise the dynamics of a system when data exhibits deterministic chaos resulting in a continuous spectrum (as opposed to a set of harmonics with a simple relationship). This is discussed by F C Moon in a work entitled "Chaotic Vibrations", Wiley-Interscience. Spectrally selective signal processing of this kind, intended to simplify analysis, can render the apparent system behaviour more complex.
The behaviour of a dynamical system is commonly represented by a curve in a multi-dimensional phase space. Successive points on the curve indicate the evolution of the system as a function of time. The line is referred to as a "trajectory", and the region of phase space to which it is confined after an arbitrary time is called an "attractor". If the dynamical system is chaotic, the region is called a "strange attractor". The attractor is the phase space region in which the system behaviour is localised, and to which it can be said to be "attracted".
There is a need for a device which is applicable to the analysis of nonlinear systems, since traditional techniques merely complicate the apparent behaviour instead of reducing it to identifiable components. Traditional techniques Such as Fourier spectral analysis are applicable only to systems which can be modelled as the linear superposition of a limited number of harmonic modes.
There is a particular need for a device which can detect a change from normal system behaviour to chaotic behaviour. This is of particular importance in the field of aero-engines and other mechanical motive power sources. Such power sources are characterised by highly regular oscillatory and/or rotational behaviour in normal operation, but develop irregular characteristics before failing catastrophically. A device capable of detecting onset or wear-related irregular behaviour would provide a means for anticipating and avoiding catastrophic failure by prior shut-down and repair. It might also provide for reduced servicing costs, since maintenance could be deferred until system behaviour warranted it.